Why is the radius $\sqrt{a}$ and not $a$?

Great catch! The standard equation of a circle is $x^2+y^2=r^2$. Comparing this to $x^2+y^2=a$, we see that $r^2 = a$, which means $r = \sqrt{a}$. Always be careful with the constant term in circle equations!

JEE Main 2021 (31 Aug Shift 2)

LEVELJEE Main

Animated Solution for Mathematics - Conic Sections: A tangent line $L$ is drawn at the point $(2, - 4)$ on the parabola $y^{2} = 8 x$ . If the line $L$ is also tangent to the circle $x^{2} + y^{2} = a$ , then 'a' is equal to ,

Q: Wait, why do we use $yy_1 = 2a(x+x_1)$ instead of finding the slope using differentiation?

You absolutely can use differentiation! Finding the slope $m = \frac{dy}{dx}$ at $(2, -4)$ is a perfectly valid path. However, the formula $yy_1 = 2a(x+x_1)$ is a powerful shortcut in coordinate geometry that saves time and reduces the chance of calculation errors. In JEE, speed and accuracy are your best friends.

Enter Numerical Value:

Progressive Hints

Hint 1

Visualize the geometry: You have a line

L

that acts as a bridge. It is born from the parabola at a specific point and then must satisfy the strict geometric requirement of being tangent to a circle centered at the origin. Don't try to solve for everything at once; focus on the line first.

Hint 2

Recall your toolkit: For the parabola

y^{2} = 4 a x

, the tangent at

(x_{1}, y_{1})

is given by

y y_{1} = 2 a (x + x_{1})

. Once you have the equation of the line

A x + B y + C = 0

, remember that for it to be tangent to a circle

x^{2} + y^{2} = r^{2}

, the perpendicular distance from the center

(0, 0)

to the line must equal the radius

r

Hint 3

Execution: First, determine the equation of the tangent line

L

by substituting the point

(2, - 4)

into the tangent formula for

y^{2} = 8 x

. Once you have the line in the form

x + y + 2 = 0

, set the perpendicular distance from

(0, 0)

equal to

a

and solve for

a

Visualized Solution

Visualize the Parabola and Point

Given Parabola: $y^{2} = 8 x$
Point of Tangency: $P (2, - 4)$

Tangent Formula for $y^{2} = 4 a x$

Equation of tangent to $y^{2} = 4 a x$ at $(x_{1}, y_{1})$ is:
$y y_{1} = 2 a (x + x_{1})$

Substitute Point $(2, - 4)$

Compare $y^{2} = 8 x$ with $y^{2} = 4 a x$ to get $2 a = 4$ .
Substitute $x_{1} = 2$ , $y_{1} = - 4$ :
$- 4 y = 4 (x + 2)$

Simplify Tangent Equation

Divide by $4$ :
$- y = x + 2$
Rearrange to standard form $A x + B y + C = 0$ :
$x + y + 2 = 0$

Introduce the Circle

Circle equation: $x^{2} + y^{2} = a$
Center: $C (0, 0)$
Radius: $r = a$

Condition for Tangency

Condition for tangency:
Perpendicular distance from center to line equals radius.
Distance formula: $d = \frac{∣ A x _{0} + B y _{0} + C ∣}{A ^{2} + B ^{2}}$

Calculate Distance from Origin

Center $(0, 0)$ , Line $x + y + 2 = 0$
$d = \frac{∣1 ( 0 ) + 1 ( 0 ) + 2∣}{1 ^{2} + 1 ^{2}}$

Simplify the Distance

$d = \frac{2}{2}$
$d = 2$

Solve for 'a'

Equate distance to radius:
$a = 2$
Square both sides:
$a = 2$

Final Takeaway

Key Takeaway:
Tangent to $y^{2} = 4 a x$ at $(x_{1}, y_{1})$ is $y y_{1} = 2 a (x + x_{1})$ .
A line is tangent to a circle if its distance from the center equals the radius.
Final Answer: $a = 2$

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The Sigma Insight: Equation of Tangent and Normal

Solution Diagram

Analyzing the Setup

The parabola is defined by the equation

y^{2} = 8 x

. By comparing this to the standard form

y^{2} = 4 A x

(where

A

is the parabola parameter), we identify

4 A = 8

, which gives

A = 2

We are given the point

P (2, - 4)

on this parabola. This point serves as the anchor for our tangent line

L

The Tangent's Identity

To find the equation of the tangent line

L

at point

(x_{1}, y_{1})

, we use the standard formula for a parabola:

y y_{1} = 2 A (x + x_{1})

Substituting our known values

A = 2

x_{1} = 2

, and

y_{1} = - 4

, we obtain:

y (- 4) = 2 (2) (x + 2)

Simplifying this expression:

- 4 y = 4 (x + 2)

- y = x + 2

x + y + 2 = 0

Thus, the equation of our tangent line

L

x + y + 2 = 0

The Circle's Embrace

We consider the circle

x^{2} + y^{2} = a

. This circle is centered at the origin

(0, 0)

with a radius

r = a

For the line

L

to be tangent to this circle, the perpendicular distance

d

from the center

(0, 0)

to the line

x + y + 2 = 0

must equal the radius

r

. We use the distance formula:

d = \frac{∣ A x _{0} + B y _{0} + C ∣}{A ^{2} + B ^{2}}

Substituting the center

(0, 0)

and the line coefficients into the formula:

d = \frac{∣1 ( 0 ) + 1 ( 0 ) + 2∣}{1 ^{2} + 1 ^{2}}

d = \frac{2}{2} = 2

Final Calculation

Since the line is tangent to the circle, the distance

d

must equal the radius

r

. We have established that

d = 2

and

r = a

Equating these two values:

a = 2

Squaring both sides, we find the final value: $a = 2$

Common Traps & FAQs

Wait, why do we use

y y_{1} = 2 a (x + x_{1})

instead of finding the slope using differentiation?

You absolutely can use differentiation! Finding the slope

m = \frac{d y}{d x}

(2, - 4)

is a perfectly valid path. However, the formula

y y_{1} = 2 a (x + x_{1})

is a powerful shortcut in coordinate geometry that saves time and reduces the chance of calculation errors. In JEE, speed and accuracy are your best friends.

How do I know if the point

(2, - 4)

is actually on the parabola?

Always verify! Before starting, plug the coordinates into the equation.

(- 4)^{2} = 16

and

8 (2) = 16

. Since

16 = 16

, the point is confirmed. It's a small habit, but it prevents catastrophic errors later.

Why is the radius

a

and not

a

Great catch! The standard equation of a circle is

x^{2} + y^{2} = r^{2}

. Comparing this to

x^{2} + y^{2} = a

, we see that

r^{2} = a

, which means

r = a

. Always be careful with the constant term in circle equations!

Animated Solution for Mathematics - Conic Sections: A tangent line $L$ is drawn at the point $(2, - 4)$ on the parabola $y^{2} = 8 x$ . If the line $L$ is also tangent to the circle $x^{2} + y^{2} = a$ , then 'a' is equal to ,

Enter Numerical Value:

Visualized Solution

Visualize the Parabola and Point

Tangent Formula for $y^{2} = 4 a x$

Substitute Point $(2, - 4)$

Simplify Tangent Equation

Introduce the Circle

Condition for Tangency

Calculate Distance from Origin

Simplify the Distance

Solve for 'a'

Final Takeaway

The Sigma Insight: Equation of Tangent and Normal

Analyzing the Setup

The Tangent's Identity

The Circle's Embrace

Final Calculation

Similar Questions

The equation of a tangent to the parabola $y^{2} = 8 x$ is $y = x + 2$ . The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is

If the common tangent to the parabolas, $y^{2} = 4 x$ and $x^{2} = 4 y$ also touches the circle, $x^{2} + y^{2} = c^{2}$ , then $c$ is equal to:

Let $L$ be a tangent line to the parabola $y^{2} = 4 x - 20$ at $(6, 2)$ . If $L$ is also a tangent to the ellipse $\frac{x ^{2}}{2} + \frac{y ^{2}}{b} = 1$ , then the value of $b$ is equal to :

If one end of focal chord $A B$ of the parabola $y^{2} = 8 x$ is at $A (1/2, - 2)$ , then the equation of tangent to it at $B$ is

If the tangent to the parabola $y^{2} = x$ at a point $(α, β), (β > 0)$ is also a tangent to the ellipse, $x^{2} + 2 y^{2} = 1$ , then $α$ is equal to :

If one end of a focal chord $A B$ of the parabola $y^{2} = 8 x$ is at $A (\frac{1}{2}, - 2)$ , then the equation of the tangent to it at $B$ is :

Let $x^{2} + y^{2} + A x + B y + C = 0$ be a circle passing through $(0, 6)$ and touching the parabola $y = x^{2}$ at $(2, 4)$ . Then $A + C$ is equal to ____.

If the tangents on the ellipse $4 x^{2} + y^{2} = 8$ at the points $(1, 2)$ and $(a, b)$ are perpendicular to each other, then $a^{2}$ is equal to :

If the normals of the parabola $y^{2} = 4 x$ drawn at the end points of its latus rectum are tangents to the circle $(x - 3)^{2} + (y + 2)^{2} = r^{2}$ , then the value of $r^{2}$ is

Let P be a point on the parabola, $x^{2} = 4 y$ . If the distance of P from the centre of the circle, $x^{2} + y^{2} + 6 x + 8 = 0$ is minimum, then the equation of the tangent to the parabola at P, is :

Enter Numerical Value:

Visualized Solution

Visualize the Parabola and Point

Tangent Formula for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } y2=4ax

Substitute Point .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (2,−4)

Simplify Tangent Equation

Introduce the Circle

Condition for Tangency

Calculate Distance from Origin

Simplify the Distance

Solve for 'a'

Final Takeaway

The Sigma Insight: Equation of Tangent and Normal

Analyzing the Setup

The Tangent's Identity

The Circle's Embrace

Final Calculation

Similar Questions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation of a tangent to the parabola y2=8x is y=x+2. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the common tangent to the parabolas, y2=4x and x2=4y also touches the circle, x2+y2=c2, then c is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let L be a tangent line to the parabola y2=4x−20 at (6,2). If L is also a tangent to the ellipse 2x2​+by2​=1, then the value of b is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If one end of focal chord AB of the parabola y2=8x is at A(1/2,−2), then the equation of tangent to it at B is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the tangent to the parabola y2=x at a point (α,β),(β>0) is also a tangent to the ellipse, x2+2y2=1, then α is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If one end of a focal chord AB of the parabola y2=8x is at A(21​,−2), then the equation of the tangent to it at B is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let x2+y2+Ax+By+C=0 be a circle passing through (0,6) and touching the parabola y=x2 at (2,4). Then A+C is equal to ____.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the tangents on the ellipse 4x2+y2=8 at the points (1,2) and (a,b) are perpendicular to each other, then a2 is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the normals of the parabola y2=4x drawn at the end points of its latus rectum are tangents to the circle (x−3)2+(y+2)2=r2, then the value of r2 is

Tangent Formula for $y^{2} = 4 a x$

Substitute Point $(2, - 4)$

The equation of a tangent to the parabola $y^{2} = 8 x$ is $y = x + 2$ . The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is

If the common tangent to the parabolas, $y^{2} = 4 x$ and $x^{2} = 4 y$ also touches the circle, $x^{2} + y^{2} = c^{2}$ , then $c$ is equal to:

Let $L$ be a tangent line to the parabola $y^{2} = 4 x - 20$ at $(6, 2)$ . If $L$ is also a tangent to the ellipse $\frac{x ^{2}}{2} + \frac{y ^{2}}{b} = 1$ , then the value of $b$ is equal to :

If one end of focal chord $A B$ of the parabola $y^{2} = 8 x$ is at $A (1/2, - 2)$ , then the equation of tangent to it at $B$ is

If the tangent to the parabola $y^{2} = x$ at a point $(α, β), (β > 0)$ is also a tangent to the ellipse, $x^{2} + 2 y^{2} = 1$ , then $α$ is equal to :

If one end of a focal chord $A B$ of the parabola $y^{2} = 8 x$ is at $A (\frac{1}{2}, - 2)$ , then the equation of the tangent to it at $B$ is :

Let $x^{2} + y^{2} + A x + B y + C = 0$ be a circle passing through $(0, 6)$ and touching the parabola $y = x^{2}$ at $(2, 4)$ . Then $A + C$ is equal to ____.

If the tangents on the ellipse $4 x^{2} + y^{2} = 8$ at the points $(1, 2)$ and $(a, b)$ are perpendicular to each other, then $a^{2}$ is equal to :

If the normals of the parabola $y^{2} = 4 x$ drawn at the end points of its latus rectum are tangents to the circle $(x - 3)^{2} + (y + 2)^{2} = r^{2}$ , then the value of $r^{2}$ is